Polynomials can be divided just like numeric constants, either by factoring or by long division. The method you use will depend on how complex the dividend and divisor of the polynomial are.
Method 1 of 3: Determine Which Approach to Use
Step 1. Observe how complex the divisor is
The complexity of the divisor (the polynomial you will divide by) compared to the dividend (the polynomial you will divide by the divisor) will determine which is the best approximation.
- If the divisor is a monomial (a single-term polynomial), either a variable with a coefficient, or a constant (a number without a variable next to it), you can probably factor the dividend and cancel one of the resulting factors with the divider. See "Factoring the Dividend" for more instructions and examples.
- If the divisor is a binomial (two-term polynomial), perhaps you can factor the dividend and cancel one of the resulting factors with the divisor.
- If the divisor is a trinomial (three-term polynomial), you may be able to factor both the dividend and divisor, cancel the common factor, and then continue factoring the dividend or use long division.
- If the divisor is a polynomial with more than three factors, you will probably have to use long division. See "Using Division of Long Polynomials" for more instructions and examples.
Step 2. See how complex the dividend is
If by looking at the dividing polynomial you can't figure out whether you should try to factor the dividend, then look directly at the dividend.
- If the dividend has three terms or less, you can probably factor it out and cancel it with the divisor.
- If the dividend has more than three terms, you should probably divide it by the divisor using long division.
Method 2 of 3: Factor the dividend
Step 1. Observe and try to find out if all the terms in the dividend contain a factor common to the divisor
If so, you can factor it and you can probably cancel it with the divisor.
- If you must divide the binomial 3x - 9 by 3, you can factor the 3 into both terms of the binomial, making it 3 (x - 3). Then you can cancel the divisor of 3, leaving a quotient of x - 3.
- If you must divide the binomial 24x3 - 18x2 times 6x, you can factor 6x into both terms of the binomial, turning it into 6x (4x2 - 3). You can cancel the divisor of 6x, leaving a quotient of 4x2 - 3.
Step 2. Look for special patterns in the dividend that tell you that you can factor it
Certain polynomials have terms that tell you that you can factor them. If one of those terms matches the divisor, you can cancel them, leaving the resulting factor as a quotient. Here are some patterns to look for:
- Difference of perfect squares. This is a binomial with the form ‘’ a2x2 - b '', where the values of '' a2’’ And ‘’ b2’’ Are perfect squares. These binomials can be factored into two binomials (ax + b) (ax - b), where a and b are the square roots of the coefficient and constants of the previous binomial.
- Perfect square trinomial. This trinomial has the form a2x2 + 2abx + b 2. It can be factored as (ax + b) (ax + b), which can also be expressed as (ax + b) 2. If the sign in front of the second term is a minus sign, the factors of the binomial will have the form (ax - b) (ax - b).
- Sum or difference of cubes. This is a binomial with the form a3x3 + b3 hear3x3 - b3, where the values from ‘’ to3’’ And ‘’ b3’’ Are perfect cubes. These binomials can be factored into a binomial and a trinomial. A sum of cubes is factored to (ax + b) (a2x2 - abx + b2). A difference of cubes is factored to (ax - b) (a2x2 + abx + b2).
Step 3. Use trial and error to factor the dividend
If you can't figure out a distinguishable pattern in the dividend to figure out how to factor it, you can try several possible factoring combinations. You can do this by first looking at the constant and trying to find several factors for it, and then the coefficient for the middle term.
- For example, if the dividend is x2 - 3x - 10, you could look at the factors of 10 and use 3 to determine which of the factor pairs is correct.
- The number 10 can be factored into factors of 1 and 10, or 2 and 5. Because the sign in front of it is negative, one of the binomials in the factorization must have a negative number in front of its constant.
- The number 3 is the difference between 2 and 5, therefore these must be the constants of the binomials of the factorization. Because the sign in front of 3 is negative, the binomial that 5 has must be the one with a negative sign. The factors of the binomial will therefore be (x - 5) (x + 2). If the divisor is one of these factors, you can cancel them, and the resulting factor will be the quotient.
Method 3 of 3: Use long division of polynomials
Step 1. State the division
You should write long division of polynomials in the same way that you do to divide numbers. The dividend goes below the long division bar, while the divisor goes to the left.
- If you must divide x2 + 11 x + 10 times x +1, so x2 + 11 x + 10 goes under the bar, while x + 1 goes to the left.
Step 2. Divide the first term of the dividend by the first term of the divisor
The result of this division is written above the division bar.
- In our example, dividing x2, the first term of the dividend, times x, the first term of the divisor, gives x. You must write an x above the division bar, above the x2.
Step 3. Multiply the x in the quotient position by the divisor
Write the result of the multiplication under the leftmost terms of the dividend.
- Continuing with our example, multiplying x + 1 by x produces the result x2 + x. You should write this under the first two terms of the dividend.
Step 4. Subtract it from the dividend
To do this, first reverse the signs of the product of the multiplication. After subtracting, drag down the remaining terms of the dividend.
- By reversing the signs of x2 + x you will have –x2 - x. If you subtract this from the first two terms of the dividend, you will get 10x. After dragging down the remaining terms of the dividend, you will have 10x + 10 as a provisional quotient to continue with the division process.
Step 5. Repeat the previous three steps for the provisional quotient
Once again, you must divide the provisional quotient by the first term of the divisor, write that result above the division bar, after the first term of the quotient, multiply the result by the divisor, and then calculate what to subtract from it. to the provisional quotient.
- Since x goes 10 times in 10 x, you must write “+ 10” after x in the corresponding quotient position on the division bar.
- Multiplying x + 1 by 10 results in 10x + 10. You must write this below the provisional quotient and reverse the signs to do the subtraction, which will transform it into -10x - 10.
- When you do the subtraction, the remainder you will have will be 0. Therefore, divide x2 + 11 x + 10 times x +1 produces a quotient of x + 10. (You could have obtained the same result by factoring, but this example was chosen to make the division fairly simple.)
- If, when you perform long divisions of polynomials, you have a remainder that is not zero, you can make that remainder part of the quotient, writing it as a fraction using the remainder as the numerator and the divisor as the denominator. If, in our long division example, the dividend had been x2 + 11 x + 12 instead of x2 + 11 x + 10, by dividing the dividend by x + 1 you would have obtained a remainder of 2. The complete quotient, therefore, should be expressed as follows: x + 10 + 2 / (x + 1).
- If your dividend has a gap in the degrees of its terms, such as 3x3+ 9x2+18, you can insert the missing term with a coefficient of 0, in this case 0x to make it easier to locate the other terms during division. Doing this will not change the value of the dividend.
- Note that some algebra books format polynomial long divisions with the right-justified quotient and dividend, or with terms presented so that terms of the same degree within a polynomial line up with each other. However, it will probably be easier for you, when doing the division by hand, to left justify the quotient and dividend as indicated in the previous steps.
- Keep your columns aligned when doing long divisions of polynomials to avoid incorrectly subtracting the terms of each one.
- When writing the quotient of a division of polynomials that includes a fractional term, always use a plus sign between the integer term (or the integer variable) and the fractional term.